Abstract
The multifractal analysis of stochastic processes deals with the fine scale
properties of the sample paths and seeks for some global scaling property that would
enable extracting the so-called spectrum of singularities. In this paper we establish bounds
on the support of the spectrum of singularities. To do this, we prove a theorem that
complements the famous Kolmogorov's continuity criterion. The nature of these bounds
helps us identify the quantities truly responsible for the support of the spectrum. We then
make several conclusions from this. First, specifying global scaling in terms of moments
is incomplete due to possible infinite moments, both of positive and negative order. The
divergence of negative order moments does not affect the spectrum in general. On the
other hand, infinite positive order moments make the spectrum of self-similar processes
nontrivial. In particular, we show that the self-similar stationary increments process
with the nontrivial spectrum must be heavy-tailed. This shows that for determining the
spectrum it is crucial to capture the divergence of moments. We show that the partition
function is capable of doing this and also propose a robust variant of this method for
negative order moments.
Item Type: |
Article
|
Date Type: |
Publication |
Status: |
Published |
Schools: |
Mathematics |
Publisher: |
World Scientific Publishing |
ISSN: |
0218-348X |
Date of First Compliant Deposit: |
3 April 2018 |
Date of Acceptance: |
28 March 2018 |
Last Modified: |
19 Oct 2019 13:09 |
URI: |
http://orca.cf.ac.uk/id/eprint/110422 |
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